1. Membrane Trafficking: a) Membrane wave: In collaboration with Dr. Min Wu from National University of Singapore, we established a mechanochemical feedback model that accounts for the ultrafast rhythmic propagation (>1micron/sec) of the endocytic machine on plasma membrane in immune cells. We demonstrated that the feedbacks between membrane shape sensing and sculpturing by endocytic machine underlie the oscillations in both the cortical protein levels and the local membrane shape changes. In the wake of these oscillations, phase wave emerges from the global activation profile of the cortical protein recruitment. Instead of physically moving in space, it is the spatial gradient in the timings of the local oscillation that gives the appearance of propagation along the membrane. The phase velocity, inversely proportional to such spatial gradient, dictates the apparent ultrafast propagation that is much faster than any conventional cortex-bound rhythmic propagation. An even larger membrane deformation elicits the Ca2+ pathway that synchronizes the rhythms into a coherent standing oscillation across the entire membrane surface. Plasma membrane in secretory cells is thus an excitable medium: the membrane shape change supports the ultrafast long-ranged signal transmission across cortex. This paper is currently under review in PLoS Biology. 2. Cell Division: a) Spatial-temporal regulation of spindle assembly checkpoint: Faithful chromosome segregation in mitosis requires stable microtubule spindle attachment at the kinetochores (KT) of each chromosome. Until then the spindle assembly checkpoint (SAC) is active to prevent mitotic progression. However, the everlasting stochastic fluctuations and large KT number in the cell would deny robust timing of SAC silencing. From the stably attached KT, SAC components stream toward the spindle poles (SP). Incorporating the spatial-temporal regulation, we established a theoretical model that unprecedentedly accounted for the fidelity of SAC silencing. The poleward streaming from the attached KTs is integrated by the SP, yet diverted by the unattached KTs until the last KT-spindle attachment, causing a >2 fold jump in the SP accumulation. Such jump robustly triggers SAC silencing after and only after the last KT-spindle attachment. Our model explained intriguing observations on mitosis and offered a unified conceptual framework: Spatial-temporal regulation ensures the fidelity of SAC silencing. This paper is currently under review in Nature Communications. b) The role of spindle pole organization in faithful mitotic exit: In previous work, we found that the spindle pole integrates the information from the kinetochores to govern the progression of mitosis, in particular the SAC silencing with regard to the last kinetochore-spindle attachment and thereby the correct timing for anaphase onset. In this study, we extended our model to incorporate the effects of supernumerary and the disorganization of spindle poles, which often manifest themselves in cancer cells. We found that fine-control over the number, the geometry and the size of spindle poles is the fundamental determinant for mitotic progression. Furthermore, the organization of the spindle poles must coordinate with the kinetochore tension for faithful anaphase onset. This result sheds light on the origin and the treatment of cancer cells. It puts diverse observations in cancer cell mitosis into perspective. The draft is currently in preparation. 3. Cell Motility: a) Mechanochemistry of focal adhesion formation: Durotaxis cells prefer to migrate toward stiffer substrate is important for many physiological processes. Focal adhesion (FA) is a dynamically formed organelle, serving as the foot of migrating cells. To better understand the mechanosensation underlying durotaxis, we provided the first theoretical model that integrates the contributions of branched actin network and stress fiber in the FA formation. It captured the salient features of FA growth in coupling with the cell leading edge protrusion. The model predicted two traction force peaks emerging within the growing FA: While the distal traction peak originates from the catch bonds that mediate FA-retrograde actin flux engagement, the central one is generated by the actomyosin contractility from stress fiber. The centraal traction peak oscillation due to the stress fiber-mediated negative feedback optimizes the range of FA mechanosensing on substrate stiffness. The competition between the two sources of tractions gives rise to the traction peak oscillation within single FAs. We experimentally perturbed the two types of actin networks, and convincingly verified these unique model predictions. Our study thereby established the coherent picture of FA formation. FA is truly a mechanosensory organelle: Its traction force generation is part of the FA-intrinsic regulatory feedbacks, which consolidate the dynamics of branched actin network and stress fiber to precisely measure up the substrate stiffness in the physiological range. Our work thus sheds light on the mechanistic nature of durotaxis. This paper is in collaboration with Dr. Clare Waterman's lab, and to be submitted to Nature Cell Biology. b) Nonlocal effects in governing the orientation of branched actin network: In our previous work published in Physical Biology, we carried out stochastic simulation that provides the unified framework on the load-velocity relationship of branched actin network. We found that the orientation of branched actin network is one of the key determinants for the adaptive mechanical response of the actin network. In this study, we derived the analytic model on how the cell membrane dictates actin network orientation. While many models have been developed to study the role of branching actin networks in motility, one important component of those models is the distribution of filament orientations relative to the cell membrane. Two mean-field models previously proposed are generalized and analyzed. In particular, we find that both models uniquely select for a dominant orientation pattern. In the linear case, the pattern is the eigenfunction associated with the principal eigenvalue. In the nonlinear case, we show there exists a unique equilibrium and that the equilibrium is locally stable. Approximate techniques are then used to provide evidence for global stability. This paper is currently under review in SIAM Journal of Applied Mathematics.